Author: UrsFormat: MarkdownItexAdded the first part of the proof that a $\mathbb{Z}$-grading on a commutative ring $R$ is the same as a $\mathbb{G}_m$-action on $Spec R$. (All a bit rough for the time being.)

Added the first part of the proof that a &Zopf;\mathbb{Z}-grading on a commutative ring RR is the same as a &Gopf;m\mathbb{G}_m-action on SpecRSpec R. (All a bit rough for the time being.)

Author: UrsFormat: MarkdownItexadded to [[projective space]] the remainder of the proof that $\mathbb{G}$-actions on $Spec R$ are equivalent to $\mathbb{Z}$-gradings of $R$.
(I guess I should move that elsewhere.)

added to projective space the remainder of the proof that &Gopf;\mathbb{G}-actions on SpecRSpec R are equivalent to &Zopf;\mathbb{Z}-gradings of RR.

Author: UrsFormat: MarkdownItexI have spelled out how real and projective space become topological manifolds and smooth manifolds, [here](https://ncatlab.org/nlab/show/projective+space#RealAndComplexProjectiveSpace).

I have spelled out how real and projective space become topological manifolds and smooth manifolds, here.

Author: UrsFormat: MarkdownItexI have filled in full details in the proof of the CW-structure at _[[complex projective space]]_. Then I copied this over also to _[Projective space -- Examples -- Real and complex projective space](https://ncatlab.org/nlab/show/projective+space#RealAndComplexProjectiveSpace)_, so that the discussion there is now a self-contained proof of the manifold structure on real/complex projective space.

Author: UrsFormat: MarkdownItexAmong the list of elementary facts, the statement that $S^n \to \mathbb{R}P^n$ is locally trivial had been missing. For completeness, have now included statement and proof [here](https://ncatlab.org/nlab/show/projective+space#nSphereAsCoveringSpaceOverRealProjectiveSpace).

Among the list of elementary facts, the statement that Sn&rightarrow;&Ropf;PnS^n \to \mathbb{R}P^n is locally trivial had been missing. For completeness, have now included statement and proof here.